When I want to reach to a certain part of a word in Vim (horizontally) I simply press w, b, f or t (something). Is there a good way of doing this vertically? For instance, I want to get to a certain word or a letter of a line that is below or above the line I'm currently in.

you can type a number and then use the up/down arrow to move up/down that many lines

small g jumps to the first line of the document

capital G jumps to the last line of the document

etc.

there are a gazillion of other different ways, vim is very flexible in this respect. I would recommend to get a vim cheat sheet first for basic commands, and then as you have time, go through the vim documentation on this subject.

I jump around by searching for unique strings where I want to go. In Vim, you can do this with:

/search-string

I usually scroll forwards and backwards using n and N once I've entered a search.

I also scroll using Ctrl-F and Ctrl-B to page through code quickly. But I use the search trick most often. Sometimes you can get lost when going through search results with n and N. A good command for orienting yourself is Ctrl-G, which prints out something like this at the bottom of the Vim screen:

as previously mentioned H,M,L(as in high, middle, low)
lower case h j k l are the typical console game commands for left, down, up, right and work the same in vi (j,k work in gmail if you enable key commands)